How to make "object A" keep distance from "vehicle with physics" and follow path at the same time?
I suppose it kind of depends on how you're going to animate the car.  But one way would be to use a boolean.

If you think about, you're trying to find the intersection of a sphere centered on the car and the curve.  So start by making a (non-rendering) mesh for the curve, just a strip of planes with a curve modifier, then make a (non-rendering) sphere around the car:



Parent your sphere to your car, give it a solidify modifier, then give it a boolean/intersection modifier with your curve mesh and now you have an arbitrarily small piece of mesh that represents the intersection of the curve and the sphere:



Now you can create an empty (or a bone, or anything else), parent it to the car, and use a shrinkwrap constraint targeting the non-rendering sphere:



So far, so good.  But of course our sphere actually intersects our curve at two points, so let's delete half of it.  And let's subdivide our curve-modified path and make it more narrow to reduce our error.



Look, your car doesn't even have to stay on the ground.

One potential problem here is if you have a very windy curve, there may be multiple points of intersection.  You can always adjust the size of your sphere as necessary to improve handling on windy roads.